On a homotopy equivalence between the 2-local geometry and the Bouc complex for the sporadic group McL

Abstract

We study the homotopy relation between the standard 2-local geometry ∆ and the Bouc complex for the sporadic finite simple group McL. Mathematics Subject Classification (2000). 20G05, 20C20, 51D20.

New Collections ofp-Subgroups and Homology Decompositions for Classifying Spaces of Finite Groups

“…Benson uses this subcollection for p = 2 and G = Co 3 , in which case it is homotopy equivalent to a 2-local geometry first mentioned by Ronan and Stroth (1984). Although this geometry is not homotopy equivalent to the Bouc collection, in Maginnis and Onofrei (2007) we prove that is homotopy equivalent to 2 Co 3 , the collection of 2-radical subgroups which contain a central involution lying in the center of some Sylow 2-subgroup. This result led us to the definition of distinguished collections of p-subgroups.…”

“…This group satisfies both conditions and . We proved in Maginnis and Onofrei (2007) that 2 G is not homotopy equivalent to 2 G or to cen 2 G .…”